from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,9,40]))
pari: [g,chi] = znchar(Mod(1033,1575))
Basic properties
Modulus: | \(1575\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1575.eo
\(\chi_{1575}(58,\cdot)\) \(\chi_{1575}(88,\cdot)\) \(\chi_{1575}(247,\cdot)\) \(\chi_{1575}(277,\cdot)\) \(\chi_{1575}(373,\cdot)\) \(\chi_{1575}(403,\cdot)\) \(\chi_{1575}(562,\cdot)\) \(\chi_{1575}(592,\cdot)\) \(\chi_{1575}(688,\cdot)\) \(\chi_{1575}(877,\cdot)\) \(\chi_{1575}(1003,\cdot)\) \(\chi_{1575}(1033,\cdot)\) \(\chi_{1575}(1192,\cdot)\) \(\chi_{1575}(1222,\cdot)\) \(\chi_{1575}(1348,\cdot)\) \(\chi_{1575}(1537,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1226,127,451)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 1575 }(1033, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)